diff --git a/lib/matplotlib/backends/_backend_gtk.py b/lib/matplotlib/backends/_backend_gtk.py
index 2b095c3bc7a9..2652e95d9519 100644
--- a/lib/matplotlib/backends/_backend_gtk.py
+++ b/lib/matplotlib/backends/_backend_gtk.py
@@ -5,10 +5,9 @@
 import logging
 
 import matplotlib as mpl
-from matplotlib import backend_tools, cbook
-from matplotlib.backend_bases import (
-    _Backend, NavigationToolbar2, TimerBase,
-)
+from matplotlib import _api, backend_tools, cbook
+from matplotlib.backend_bases import _Backend, NavigationToolbar2, TimerBase
+from matplotlib.backend_tools import Cursors
 
 # The GTK3/GTK4 backends will have already called `gi.require_version` to set
 # the desired GTK.
@@ -62,6 +61,18 @@ def _create_application():
     return _application
 
 
+def mpl_to_gtk_cursor_name(mpl_cursor):
+    return _api.check_getitem({
+        Cursors.MOVE: "move",
+        Cursors.HAND: "pointer",
+        Cursors.POINTER: "default",
+        Cursors.SELECT_REGION: "crosshair",
+        Cursors.WAIT: "wait",
+        Cursors.RESIZE_HORIZONTAL: "ew-resize",
+        Cursors.RESIZE_VERTICAL: "ns-resize",
+    }, cursor=mpl_cursor)
+
+
 class TimerGTK(TimerBase):
     """Subclass of `.TimerBase` using GTK timer events."""
 
diff --git a/lib/matplotlib/backends/backend_gtk3.py b/lib/matplotlib/backends/backend_gtk3.py
index 1bd8b7334eca..dbbb4c375e1c 100644
--- a/lib/matplotlib/backends/backend_gtk3.py
+++ b/lib/matplotlib/backends/backend_gtk3.py
@@ -8,11 +8,9 @@
 from matplotlib import _api, backend_tools, cbook
 from matplotlib._pylab_helpers import Gcf
 from matplotlib.backend_bases import (
-    _Backend, FigureCanvasBase, FigureManagerBase, NavigationToolbar2,
-    TimerBase, ToolContainerBase)
+    FigureCanvasBase, FigureManagerBase, ToolContainerBase)
 from matplotlib.backend_tools import Cursors
 from matplotlib.figure import Figure
-from matplotlib.widgets import SubplotTool
 
 try:
     import gi
@@ -31,7 +29,6 @@
 from gi.repository import Gio, GLib, GObject, Gtk, Gdk
 from . import _backend_gtk
 from ._backend_gtk import (
-    _create_application, _shutdown_application,
     backend_version, _BackendGTK, _NavigationToolbar2GTK,
     TimerGTK as TimerGTK3,
 )
@@ -68,16 +65,9 @@ def cursord(self):
 
 @functools.lru_cache()
 def _mpl_to_gtk_cursor(mpl_cursor):
-    name = _api.check_getitem({
-        Cursors.MOVE: "move",
-        Cursors.HAND: "pointer",
-        Cursors.POINTER: "default",
-        Cursors.SELECT_REGION: "crosshair",
-        Cursors.WAIT: "wait",
-        Cursors.RESIZE_HORIZONTAL: "ew-resize",
-        Cursors.RESIZE_VERTICAL: "ns-resize",
-    }, cursor=mpl_cursor)
-    return Gdk.Cursor.new_from_name(Gdk.Display.get_default(), name)
+    return Gdk.Cursor.new_from_name(
+        Gdk.Display.get_default(),
+        _backend_gtk.mpl_to_gtk_cursor_name(mpl_cursor))
 
 
 class FigureCanvasGTK3(Gtk.DrawingArea, FigureCanvasBase):
@@ -317,10 +307,10 @@ class FigureManagerGTK3(FigureManagerBase):
         The Gtk.VBox containing the canvas and toolbar
     window : Gtk.Window
         The Gtk.Window
-
     """
+
     def __init__(self, canvas, num):
-        app = _create_application()
+        app = _backend_gtk._create_application()
         self.window = Gtk.Window()
         app.add_window(self.window)
         super().__init__(canvas, num)
@@ -481,7 +471,7 @@ def __init__(self, canvas, window):
 
         self.show_all()
 
-        NavigationToolbar2.__init__(self, canvas)
+        _NavigationToolbar2GTK.__init__(self, canvas)
 
     def save_figure(self, *args):
         dialog = Gtk.FileChooserDialog(
@@ -747,7 +737,7 @@ def error_msg_gtk(msg, parent=None):
     FigureCanvasGTK3, _backend_gtk.RubberbandGTK)
 
 
-@_Backend.export
+@_BackendGTK.export
 class _BackendGTK3(_BackendGTK):
     FigureCanvas = FigureCanvasGTK3
     FigureManager = FigureManagerGTK3
diff --git a/lib/matplotlib/backends/backend_gtk4.py b/lib/matplotlib/backends/backend_gtk4.py
index 74c1317e4eb8..7471bf7fb669 100644
--- a/lib/matplotlib/backends/backend_gtk4.py
+++ b/lib/matplotlib/backends/backend_gtk4.py
@@ -2,17 +2,12 @@
 import io
 import os
 from pathlib import Path
-import sys
 
 import matplotlib as mpl
 from matplotlib import _api, backend_tools, cbook
 from matplotlib._pylab_helpers import Gcf
 from matplotlib.backend_bases import (
-    _Backend, FigureCanvasBase, FigureManagerBase, NavigationToolbar2,
-    TimerBase, ToolContainerBase)
-from matplotlib.backend_tools import Cursors
-from matplotlib.figure import Figure
-from matplotlib.widgets import SubplotTool
+    FigureCanvasBase, FigureManagerBase, ToolContainerBase)
 
 try:
     import gi
@@ -31,24 +26,11 @@
 from gi.repository import Gio, GLib, GObject, Gtk, Gdk, GdkPixbuf
 from . import _backend_gtk
 from ._backend_gtk import (
-    _create_application, _shutdown_application,
     backend_version, _BackendGTK, _NavigationToolbar2GTK,
     TimerGTK as TimerGTK4,
 )
 
 
-def _mpl_to_gtk_cursor(mpl_cursor):
-    return _api.check_getitem({
-        Cursors.MOVE: "move",
-        Cursors.HAND: "pointer",
-        Cursors.POINTER: "default",
-        Cursors.SELECT_REGION: "crosshair",
-        Cursors.WAIT: "wait",
-        Cursors.RESIZE_HORIZONTAL: "ew-resize",
-        Cursors.RESIZE_VERTICAL: "ns-resize",
-    }, cursor=mpl_cursor)
-
-
 class FigureCanvasGTK4(Gtk.DrawingArea, FigureCanvasBase):
     required_interactive_framework = "gtk4"
     supports_blit = False
@@ -108,7 +90,7 @@ def destroy(self):
 
     def set_cursor(self, cursor):
         # docstring inherited
-        self.set_cursor_from_name(_mpl_to_gtk_cursor(cursor))
+        self.set_cursor_from_name(_backend_gtk.mpl_to_gtk_cursor_name(cursor))
 
     def _mouse_event_coords(self, x, y):
         """
@@ -281,10 +263,10 @@ class FigureManagerGTK4(FigureManagerBase):
         The Gtk.VBox containing the canvas and toolbar
     window : Gtk.Window
         The Gtk.Window
-
     """
+
     def __init__(self, canvas, num):
-        app = _create_application()
+        app = _backend_gtk._create_application()
         self.window = Gtk.Window()
         app.add_window(self.window)
         super().__init__(canvas, num)
@@ -422,7 +404,7 @@ def __init__(self, canvas, window):
         self.message = Gtk.Label()
         self.append(self.message)
 
-        NavigationToolbar2.__init__(self, canvas)
+        _NavigationToolbar2GTK.__init__(self, canvas)
 
     def save_figure(self, *args):
         dialog = Gtk.FileChooserNative(
@@ -691,7 +673,7 @@ def trigger(self, *args, **kwargs):
 Toolbar = ToolbarGTK4
 
 
-@_Backend.export
+@_BackendGTK.export
 class _BackendGTK4(_BackendGTK):
     FigureCanvas = FigureCanvasGTK4
     FigureManager = FigureManagerGTK4